On the range of the derivative of a smooth mapping between Banach spaces
We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2005-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/AAA.2005.499 |
| Summary: | We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′(X)=A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance,
there exists a mapping f from ℓ1(ℕ) into ℝ2, which is bounded, Lipschitz-continuous, and so that for all x,y∈ℓ1(ℕ), if x≠y, then ‖f′(x)−f′(y)‖>1. |
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| ISSN: | 1085-3375 1687-0409 |